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A325767
Heinz numbers of integer partitions covering an initial interval of positive integers and containing their own multiset of multiplicities (as a submultiset).
1
1, 2, 12, 18, 36, 60, 120, 180, 360, 450, 540, 600, 840, 1260, 1350, 1500, 1680, 1800, 2250, 2520, 2700, 3000, 3780, 4200, 4500, 5040, 5400, 5880, 6750, 8400, 9000, 10500, 11340, 11760, 12600, 13500, 15120, 17640, 18480, 18900, 20580, 21000, 22680, 25200
OFFSET
1,2
COMMENTS
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The enumeration of these partitions by sum is given by A325766.
EXAMPLE
The sequence of terms together with their prime indices begins:
1: {}
2: {1}
12: {1,1,2}
18: {1,2,2}
36: {1,1,2,2}
60: {1,1,2,3}
120: {1,1,1,2,3}
180: {1,1,2,2,3}
360: {1,1,1,2,2,3}
450: {1,2,2,3,3}
540: {1,1,2,2,2,3}
600: {1,1,1,2,3,3}
840: {1,1,1,2,3,4}
1260: {1,1,2,2,3,4}
1350: {1,2,2,2,3,3}
1500: {1,1,2,3,3,3}
1680: {1,1,1,1,2,3,4}
1800: {1,1,1,2,2,3,3}
2250: {1,2,2,3,3,3}
2520: {1,1,1,2,2,3,4}
MATHEMATICA
red[n_]:=If[n==1, 1, Times@@Prime/@Last/@FactorInteger[n]];
Select[Range[1000], #==1||Range[PrimeNu[#]]==PrimePi/@First/@FactorInteger[#]&&Divisible[#, red[#]]&]
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 19 2019
STATUS
approved